MPPT

In the MPPT/MBPT method, once the reference CSF is chosen and the SCF orbitals belonging to this CSF are detennined, the wavefiinction T and energy E are detennined in an order-by-order maimer. The perturbation equations determine what CSFs to include and their particular order. This is one of the primary strengdis of this technique it does not require one to make fiirtlier choices, in contrast to the MCSCF and Cl treatments where one needs to choose which CSFs to include. 

As Bartlett [ ] and Pople have both demonstrated [M], there is a close relationship between the MPPT/MBPT and CC methods when the CC equations are solved iteratively starting with such an MPPT/MBPT-like initial guess for these double-excitation amplitudes. 

These approaches provide alternatives to the conventional tools of quantum chemistry. The Cl, MCSCF, MPPT/MBPT, and CC methods move beyond the single-configuration picture by adding to the wavefimction more configurations whose amplitudes they each detennine in their own way. This can lead to a very large number of CSFs in the correlated wavefimction and, as a result, a need for extraordinary computer resources. 

This can be seen by considering the second-order MPPT energy of two non-interacting Be atoms. The reference CSF is = lsi 2s- ls 2s as discussed earlier, only doubly-excited CSFs contribute to the... 

The essential features of the MPPT/MBPT approaeh are deseribed in the following artieles ... 

The MoIIer-PIesset perturbation method (MPPT) uses the single-eonfiguration SCF proeess (usually the UHF implementation) to first determine a set of LCAO-MO eoeffieients and, henee, a set of orbitals that obey F( )i = 8i (jii. Then, using an unperturbed Hamiltonian equal to the sum of these Foek operators for eaeh of the N eleetrons =... 

Ei=i N F(i), perturbation theory (see Appendix D for an introduetion to time-independent perturbation theory) is used to determine the Ci amplitudes for the CSFs. The MPPT proeedure is also referred to as the many-body perturbation theory (MBPT) method. The two names arose beeause two different sehools of physies and ehemistry developed them for somewhat different applieations. Later, workers realized that they were identieal in their working equations when the UHF H is employed as the unperturbed Hamiltonian. In this text, we will therefore refer to this approaeh as MPPT/MBPT. 

The amplitude for the so-ealled referenee CSF used in the SCF proeess is taken as unity and the other CSFs amplitudes are determined, relative to this one, by Rayleigh-Sehrodinger perturbation theory using the full N-eleetron Hamiltonian minus the sum of Foek operators H-H as the perturbation. The Slater-Condon rules are used for evaluating matrix elements of (H-H ) among these CSFs. The essential features of the MPPT/MBPT approaeh are deseribed in the following artieles J. A. Pople, R. Krishnan, H. B. Sehlegel, and J. S. Binkley, Int. J. Quantum Chem. 14, 545 (1978) R. J. Bartlett and D. M. Silver, J. Chem. Phys. 3258 (1975) R. Krishnan and J. A. Pople, Int. J. Quantum Chem. 

B. Non-Variational Methods Sueh as MPPT/MBPT and CC do not Produee Upper Bounds, but Yield Size-Extensive Energies... 

P I H I P >. It ean be shown (H. P. Kelly, Phys. Rev. 131, 684 (1963)) that this differenee allows non-variational teehniques to yield size-extensive energies. This ean be seen in the MPPT/MBPT ease by eonsidering the energy of two non-interaeting Be atoms. The referenee CSF is = Isa 2sa Isb 2sb the Slater-Condon rules limit the CSFs in P whieh ean eontribute to... 

The additivity of E and the separability of the equations determining the Cj eoeffieients make the MPPT/MBPT energy size-extensive. This property ean also be demonstrated for the Coupled-Cluster energy (see the referenees given above in Chapter 19.1.4). However, size-extensive methods have at least one serious weakness their energies do not provide upper bounds to the true energies of the system (beeause their energy funetional is not of the expeetation-value form for whieh the upper bound property has been proven). 

The implementation of the CC method begins mueh as in the MPPT/MBPT ease one seleets a referenee CSF that is used in the SCF proeess to generate a set of spin-orbitals to be used in the subsequent eorrelated ealeulation. The set of working equations of the CC teehnique given above in Chapter 19.1.4 ean be written explieitly by introdueing the form of the so-ealled eluster operator T,... 

The CC method, as presented here, suffers from the same drawbaeks as the MPPT/MBPT approaeh its energy is not an upper bound and it may not be able to aeeurately deseribe waveflinetions whieh have two or more CSFs with approximately equal amplitude. Moreover, solution of the non-linear CC equations may be diffieult and slowly (if at all) eonvergent. It has the same advantages as the MPPT/MBPT method its energy is... 

Within the CC, Cl, and MPPT/MBPT methods, one must evaluate the so-ealled responses of the Ci and Ca,i eoeffieients (3Cj/3)i)o and (3Ca,i/3/i)o that appear in the full energy response as (see above)... 

This Foek operator is used to define the unperturbed Hamiltonian of Moller-Plesset perturbation theory (MPPT) ... 

The perturbation V appropriate to this MPPT ease is the differenee between the full N-eleetronie Hamiltonian and this H ... 

The first-order MPPT wavefunction can be evaluated in terms of Slater determinants that are excited relative to the SCF reference function k. Realizing again that the perturbation coupling matrix elements I>k H i> are non-zero only for doubly excited CSF s, and denoting such doubly excited i by a,b m,n the first-order... 

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